600 ON THE CALCULATION OF THE EQUILIBRIUM AND STIFFNESS OF FRAMES. 



to produce extension -unity, will be - . We shall suppose that the effect of 



c 



pressure in producing compression of the piece is equal to that of tension in 

 producing extension, and we shall use e indifferently for extensibility and com- 

 pressibility. 



Work done against Elasticity. 



Since the extension is proportional to the force, the whole work done will 

 l>e the product of the extension and the mean value of the force ; or if x is 

 the extension and F the force, 



x = eF, 



work = %Fx = \eF* = l-x*. 



c 



When the piece is inextensible, or e = 0, then all the work applied at one end 

 is transmitted to the other, and the frame may be regarded as a machine whose 

 efficiency is perfect. Hence the following 



THEOREM. If p be the tension of the piece A due to a tension-unity 

 l>etween the points B and C, then an extension-unity taking place in A will 

 bring B and C nearer by a distance p. 



For let X be the tension and x the extension of A, Y the tension and 

 // the extension of the line BC '; then supposing all the other pieces inextensible, 

 no work will be done except in stretching A, or 



But X=pY, therefore y = px, which was to be proved. 



PROBLEM I. A tension F is applied between the points B and C of a 

 frame which is simply stiff; to find the extension of the line joining D and E, 

 all the pieces except A being inextensible, the extensibility of A being e. 



Determine the tension in each piece due to unit tension between B and C, 

 and let p be the tension in A due to this cause. 



Determine also the tension in each piece due to unit tension between D 

 and E, and let y be the tension in the piece A due to this cause. 



Then the actual tension of A is Fp, and its extension is eFp, and the 

 extension of the line DE due to this cause is Fepq by the last theorem. 



